SUq(3)SU_q(3) corepresentations and bivariate qq-Krawtchouk polynomials

The matrix elements of unitary $SU_q(3)$ corepresentations, which are analogues of the symmetric powers of the natural repesentation, are shown to be the bivariate $q$-Krawtchouk orthogonal polynomials, thus providing an algebraic interpretation of these polynomials in terms of quantum groups.Comment: 14 page

Coefficients de Clebsch-Gordan de la super-algèbre osp(1|2)

Les fonctions génératrices des coefficients de Clebsch Gordan pour la superalgèbre de Lie osp(1|2) sont dérivées en utilisant deux approches. Une première approche généralise une méthode proposée par Granovskii et Zhedanov pour l'appliquer dans le cas de osp(1|2), une algèbre dont le coproduit est torsadé. Une seconde approche repose sur la réalisation de osp(1|2) en tant qu'algèbre dynamique d'un oscillateur parabosonique et utilise une équivalence dans cette réalisation entre le changements de coordonnées polaires à cartésiennes et le problème de Clebsch-Gordan. Un chapitre moins formel précède ces dérivations et présente comment le problème de Clebsch-Gordan s'interprète en tant que réalisation d'une algèbre de fusion. La notion abstraite de fusion est introduite, soulignant son importance en physique, pour en venir au cas particulier du problème de Clebsch-Gordan. Un survol du cas de l'algèbre osp(1|2) et de ses utilisations en physique mathématique conclut ce chapitre.The generating functions for the osp(1|2) Lie superalgebra Clebsch-Gordan coefficients are derived using two approaches. The first one consists of generalizing a method first proposed by Granovskii and Zhedanov to apply it to the case of osp(1|2), an algebra with a twisted coproduct. The second one is based on the realization of the osp(1|2) as the dynamical algebra for a parabosonic oscillator and used an equivalence in this realization between a change of basis from polar to cartesian coordinates and the Clebsch-Gordan problem. A less formal chapter precedes those derivations and present how the Clebsch-Gordan problem can be interpreted as a realization of a fusion algebra. The abstract notion of fusion is introduced, mentionning its importance in physics, and leads to the particular case of the Clebsch-Gordan problem. A brief review of the problem for the osp(1|2) algebra and its uses in mathematical physics concludes this chapter

The rational Sklyanin algebra and the Wilson and para-Racah polynomials

The relation between Wilson and para-Racah polynomials and representations of the degenerate rational Sklyanin algebra is established. Second order Heun operators on quadratic grids with no diagonal terms are determined. These special or S-Heun operators lead to the rational degeneration of the Sklyanin algebra; they also entail the contiguity and structure operators of the Wilson polynomials. The finite-dimensional restriction yields a representation that acts on the para-Racah polynomials.Comment: 16 page

An Algorithmic Approach to Emergence

We suggest a quantitative and objective notion of emergence. Our proposal uses algorithmic information theory as a basis for an objective framework in which a bit string encodes observational data. Plurality of drops in the Kolmogorov structure function of such a string is seen as the hallmark of emergence. Our definition offers some theoretical results, in addition to extending the notions of coarse-graining and boundary conditions. Finally, we confront our proposal with applications to dynamical systems and thermodynamics.Comment: 39 pages, 11 figure

The Heun–Racah and Heun–Bannai–Ito algebras

5 pages, 27 ref.International audienc